A polynomial f(x) with real coefficients is said to be a sum of squares if there are polynomials p_{1}(x), p_{2}(x), \ldots, p_{n}(x) with real coefficients for which
f(x)=p_{1}^{2}(x)+p_{2}^{2}(x)+\cdots+p_{n}^{2}(x).
For example, 2 x^{4}+6 x^{2}-4 x+5 is a sum of squares because
2 x^{4}+6 x^{2}-4 x+5=\left(x^{2}\right)^{2}+\left(x^{2}+1\right)^{2}+(2 x-1)^{2}+(\sqrt{3})^{2}.
(a) Determine all values of a for which f(x)=x^{2}+4 x+a is a sum of squares.
(b) Determine all values of a for which f(x)=x^{4}+2 x^{3}+(a-7) x^{2}+(4-2 a) x+a is a sum of squares, and for such values of a, write f(x) as a sum of squares.
(c) Suppose f(x) is a sum of squares. Prove there are polynomials u(x), v(x) with real coefficients such that f(x)=u^{2}(x)+v^{2}(x).